You're planning a single performance, to play 'right first time', as a surprise.
Is it possible to learn piano without ever touching one? No! Forget it. Unless you can get regular time at a real piano, it will be a disaster. If it's not at a real piano/good keyboard (preferably with a sustain pedal), it is not practice, it's just revision or something.
Is it possible to learn to play 1 song, to performance quality, within a year? Absolutely it is... Well, performance for a solo in that situation is also as much about personality as it is skill. You can always decide not to go through with it if you don't feel ready. You will need lots of practice, regardless of what path you take. You have to be able to play this in your sleep!
A professional musician can do this within a day - given a simple enough arrangement.
Choose a song! If you don't read music, you might be better off translating it into a notation you can read more easily (invent your own if you have to). If there's a how-to-play video for your song on youtube, and you think you can do it, fantastic, start there. If there's a midi version, there is software you can use to see the notes that are played on the keyboard (e.g. http://midisheetmusic.sourceforge.net/)
There are many different ways of playing the same song. Many beginner books have 'easy-to-play' arrangements for example based on the song's vocal and a chord chart.
If you go for an easy arrangement, you can always build on it if you master it (e.g. playing chords with the right hand instead of a single-fingered melody). Whereas if you choose to learn a more complete arrangement to begin with, it might be too ambitious. The pitfall in this approach is sometimes easy-to-play arrangements are in a different key (for reasons unknown to me!).
If you have some friends that are half-decent singers, you could all be in on it, and all you'll need to do is sit at the piano and stab the right chords/rhythm whilst everybody sings along, if you choose a well-known song and get the whole room singing, you won't need to worry about your performance as much, and it will be no less of a surprise for your wife that you played piano.
If you get a teacher, make your goal clear. For example, I don't see that learning scales, sheet music, etc, is necessary for this particular goal.